Total Partial Domination in Graphs Under Some Binary Operations

Abstract

Let G = (V (G), E(G)) be a simple graph and let α ∈ (0, 1]. A set S ⊆ V (G) is
an α-partial dominating set in G if |N[S]| ≥ α |V (G)|. The smallest cardinality of an α-partial
dominating set in G is called the α-partial domination number of G, denoted by ∂α(G). An α-
partial dominating set S ⊆ V (G) is a total α-partial dominating set in G if every vertex in S is
adjacent to some vertex in S. The total α-partial domination number of G, denoted by ∂T α(G), is
the smallest cardinality of a total α-partial dominating set in G. In this paper, we characterize the
total partial dominating sets in the join, corona, lexicographic and Cartesian products of graphs
and determine the exact values or sharp bounds of the corresponding total partial domination
number of these graphs.